Subsets of vertices give Morita equivalences of Leavitt path algebras

0.17 MB·

by  Lisa Orloff Clark

SUBSETS OF VERTICES GIVE MORITA EQUIVALENCES OF
LEAVITT PATH ALGEBRAS
LISA ORLOFF CLARK, ASTRID AN HUEF, AND PAREORANGA LUITEN-APIRANA
Abstract. We show that every subset of vertices of a directed graph E gives a Morita
equivalencebetweenasubalgebraandanidealoftheassociatedLeavittpathalgebra. We
usethisobservationtoproveanalgebraicversionofatheoremofCrispandGow: certain
7
subgraphsof E canbe contractedto a new graphG such that the Leavittpath algebras
1
0 ofE andGareMoritaequivalent. Weprovideexamplestoillustratehowdesingularising
2 a graph, and in- or out-delaying of a graph, all fit into this setting.

n
a
J

1
1 1. Introduction

] Given a directed graph E, Crisp and Gow identified in [10, Theorem 3.1] a type of
A
subgraph which can be “contracted” to give a new graph G whose C∗-algebra C∗(G) is
R
Morita equivalent to C∗(E). Crisp and Gow’s construction is widely applicable, as they
.
h
point out in [10, §4]. It includes, for example, Morita equivalences of the C∗-algebras of
t
a graphs that are elementary-strong-shift-equivalent [4, 11], or are in- or out-delays of each
m
other [5].
[
The C∗-algebra of a directed graph E is the universal C∗-algebragenerated by mutually

1
orthogonal projections p and partial isometries s associated to the vertices v and edges e
v v e
8 ofE,respectively, subjecttorelations. Inparticular, therelationscapturetheconnectivity
7 of the graph. For any subset V of vertices, p converges to a projection p in the
1 Pv∈V v
multiplier algebra of C∗(E). (If V is finite, then p is in C∗(E).) Then the module pC∗(E)
3
0 implements a Morita equivalence between the corner pC∗(E)p of C∗(E) and the ideal
1. C∗(E)pC∗(E) of C∗(E). The difficult part is to identify pC∗(E)p and C∗(E)pC∗(E) with
0 known algebras. The corner pC∗(E)p may not be another graph algebra, but sometimes
7
it is (see, for example, [9]). The projection p is called full when C∗(E)pC∗(E) = C∗(E).
1
: Now let R be a commutative ring with identity. A purely algebraic analogue of the
v
i graph C∗-algebra is the Leavitt path algebra L (E) over R. This paper is based on the
X R
very simple observation that every subset V of the vertices of a directed graph E gives
r
a an algebraic version of the Morita equivalence between pC∗(E)p and C∗(E)pC∗(E) for
Leavitt pathalgebras (see Theorem 1). Weshow that thisobservation is widely applicable
by proving an algebraic version of Crisp and Gow’s theorem (see Theorem 3). A special
case of this result has been very successfully used in both [2, Section 3] and [13].
If V is infinite, we cannot make sense of the projection p in L (E), but we can make
R
sense of the algebraic analogues of the sets pC∗(E), pC∗(E)p and C∗(E)pC∗(E). For
Date: 12 January 2017.
2010 Mathematics Subject Classification. 16D70.
Key words and phrases. Directed graph, Leavitt path algebra, Morita context, Morita equivalence,
graph algebra.
This research has been supported by a University of Otago Research Grant.
1
2 L.O. CLARK,A. AN HUEF,AND P.LUITEN-APIRANA
example,
pC∗(E) = span{s s : µ,ν are paths in E and µ has range in V}
µ ν∗
has analogue
M = span {s s : µ,ν are paths in E and µ has range in V},
R µ ν∗
where we also use s and p for universal generators of L (E). Theorem 1 below gives
e v R
a surjective Morita context (M,M∗,MM∗,M∗M) between the R-subalgebra MM∗ and
the ideal M∗M of L (E). The set V is full, in the sense that M∗M = L (E), if and only
R R
if the saturated hereditary closure of V is the whole vertex set of E (see Lemma 2).
Recently, thefirst authorandSimsproved in[7, Theorem 5.1] thatequivalent groupoids
have Morita equivalent Steinberg R-algebras. They then proved that the graph groupoids
of the graphs G and E appearing in Crisp and Gow’s theorem are equivalent groupoids [7,
Proposition6.2]. SincetheSteinbergalgebraofagraphgroupoidiscanonicallyisomorphic
to the Leavitt path algebra of the graph, they deduced that the Leavitt path algebras of
L (G) and L (E) are Morita equivalent.
R R
In particular, we obtain a direct proof of [7, Proposition 6.2] using only elementary
methods. There are two advantages to our elementary approach: it illustrates on the one
hand where we have had to use different techniques from the C∗-algebraic analogue, and
on the other hand where we can just use the C∗-algebraic results already established.
2. Preliminaries
A directed graph E = (E0,E1,r,s) consists of countable sets E0 and E1, and range
and source maps r,s : E1 → E0. We think of E0 as the set of vertices, and of E1 as the
set of edges directed by r and s. A vertex v is called a infinite receiver if |r−1(v)| = ∞
and is called a source if |r−1(v)| = 0. Sources and infinite receivers are called singular
vertices.
We use the convention that a path is a sequence of edges µ = µ µ ··· such that
1 2
s(µ ) = r(µ ). We denote the ith edge in a path µ by µ . We say a path µ is finite if the
i i+1 i
sequence is finite and denote its length by |µ|. Vertices are regarded as paths of length 0.
We denote the set of finite paths by E∗ and the set of infinite paths by E∞. We usually
use the letters x,y for infinite paths. We extend the range map r to µ ∈ E∗ ∪ E∞ by
r(µ) = r(µ ); for µ ∈ E∗ we also extend the source map s by s(µ) = s(µ ).
1 |µ|
Let (E1)∗ := {e∗ : e ∈ E1} be a set of formal symbols called ghost edges. If µ ∈ E∗,
then we write µ∗ for µ∗ ...µ∗µ∗ and call it a ghost path. We extend r and s to the ghost
|µ| 2 1
paths by r(µ∗) = s(µ) and s(µ∗) = r(µ).
Let R be a commutative ring with identity and let A be an R-algebra. A Leavitt E-
family in A is a set {P ,S ,S : v ∈ E0,e ∈ E1} ⊂ A where {P : v ∈ E0} is a set of
v e e∗ v
mutually orthogonal idempotents, and
(L1) P S = S = S P and P S = S = S P for e ∈ E1;
r(e) e e e s(e) s(e) e∗ e∗ e∗ r(e)
(L2) S S = δ P for e,f ∈ E1; and
e∗ f e,f s(e)
(L3) for all non-singular v ∈ E0, P = S S .
v Pr(e)=v e e∗
For a path µ ∈ E∗ we set S := S ...S . The Leavitt path algebra L (E) is the
µ µ1 µ|µ| R
universal R-algebra generated by a universal Leavitt E-family {p ,s ,s }: that is, if A
v e e∗
is an R-algebra and {P ,S ,S } is a Leavitt E-family in A, then there exists a unique
v e e∗
MORITA EQUIVALENCES OF LEAVITT PATH ALGEBRAS 3
R-algebra homomorphism π : L (E) → A such that π(p ) = P and π(s ) = S [15, §2-3].
R v v e e
It follows from (L2) that
L (E) = span {s s∗ : µ,ν ∈ E∗}.
R R µ ν
3. Subsets of vertices of a directed graph give Morita equivalences
Theorem 1. Let E be a directed graph, let R be a commutative ring with identity and let
{p ,s ,s } be a universal generating Leavitt E-family in L (E). Let V ⊂ E0, and
v e e∗ R
M := span {s s : µ,ν ∈ E∗,r(µ) ∈ V} and M∗ := span {s s : µ,ν ∈ E∗,r(ν) ∈ V}.
R µ ν∗ R µ ν∗
Then
(1) MM∗ is an R-subalgebra of L (E);
R
(2) MM∗ = span{s s : r(µ),r(ν) ∈ V}, and M∗M is an ideal of L (E) containing
µ ν∗ R
MM∗;
(3) with actions given by multiplication in L (E), M is an MM∗–M∗M-bimodule and
R
M∗ is an M∗M–MM∗-bimodule;
(4) there are bimodule homomorphisms
Ψ: M ⊗ M∗ → MM∗ and Φ: M∗ ⊗ M → M∗M
M∗M MM∗
such that (MM∗,M∗M,M,M∗,Ψ,Φ) is a surjective Morita context.
Proof. We have
MM∗ = span {p s s s s∗p : v,w ∈ V,α,β,µ,ν ∈ E∗}.
R v µ ν∗ α β w
Products of the form s s s s∗ are either zero or of the form s s s s = s s for
µ ν∗ α β µ γ δ∗ ν∗ µγ (νδ)∗
some γ,δ ∈ E∗. Thus it is easy to see that MM∗ is a subalgebra of L (E) and
R
MM∗ = span {p s s p : v,w ∈ V,µ,ν ∈ E∗} = span{s s : µ,ν ∈ E∗,r(µ),r(ν) ∈ V}.
R v µ ν∗ w µ ν∗
Similarly, M∗M is an ideal.
To see that MM∗ ⊂ M∗M, take a spanning element s s of MM∗. Then r(µ) ∈ V,
µ ν∗
s s ∈ M, and s s = p p s s ∈ M∗M. Thus MM∗ ⊂ M∗M.
µ ν∗ µ ν∗ r(µ) r(µ)∗ µ ν∗
Since the module actions are given by multiplication in L (E) it is easy to verify
R
that M is an MM∗–M∗M-bimodule and M∗ is an M∗M–MM∗-bimodule. The function
f: M×M∗ → MM∗ defined by f(m,n) = mn is bilinear and f(md,n) = f(m,dn) for all
d ∈ M∗M. By the universal property of the balanced tensor product, there is a bimodule
homomorphism Ψ: M⊗ M∗ → MM∗ such thatΨ(m⊗n) = f(m,n) = mn. Similarly,
M∗M
there is a bimodule homomorphism Φ: M∗ ⊗ M → M∗M such that Φ(n,m) = nm.
MM∗
Both Ψ and Φ are surjective. Since multiplication in L (E) is associative, for m,m′ ∈
R
M,n,n′ ∈ M∗, we have
mΦ(n⊗m′) = mnm′ = Ψ(m⊗n)m′ and nΨ(m⊗n′) = nmn′ = Φ(n⊗m)n′.
Thus (MM∗,M∗M,M,M∗,Ψ,Φ) is a surjective Morita context. (cid:3)
In the situation of Theorem 1, we say a subset V of E0 is full if the ideal M∗M is all
of L (E). We want a graph-theoretic characterisation of fullness, and so we want an the
R
algebraic version of [5, Lemma 2.2]. We need some definitions.
4 L.O. CLARK,A. AN HUEF,AND P.LUITEN-APIRANA
For v,w ∈ E0 we write v ≤ w if there is a path µ ∈ E∗ such that s(µ) = w and
r(µ) = v. We say a subset H of E0 is hereditary if v ∈ H and v ≤ w implies w ∈ H. A
hereditary subset H of E0 is saturated if
v ∈ E0,0 < |r−1(v)| < ∞ and s(r−1(v)) ⊂ H =⇒ v ∈ H.
We denote by ΣH(V) the smallest saturated hereditary subset of E0 containing V. For
a saturated hereditary subset H of E0 we write I for the ideal of L (E) generated by
H R
{p : v ∈ H}.
v
Lemma 2. Let E be a directed graph, and let V ⊂ E0. Then V is full if and only if
ΣH(V) = E0.
Proof. Let R be a commutative ring with identity and let {p ,s ,s } be a universal
v e e∗
generating Leavitt E-family in L (E). As in Theorem 1, let M = span {s s : r(µ) ∈
R R µ ν∗
V}.
First suppose that V is full, that is, that M∗M = L (E). To see that ΣH(V) = E0,
R
fix v ∈ E0. Then p ∈ M∗M, and we can write p as a linear combination
v v
p = r s s s s
v X α,β,µ,ν α β∗ µ ν∗
(α,β)∈F1,(µ,ν)∈F2
where F ,F are finite subsets of E∗ ×E∗, and each r ∈ R and r(β) = r(µ) ∈ V.
1 2 α,β,µ,ν
Since ΣH(V) is a hereditary subset containing V, we have s(β), s(µ) ∈ ΣH(V), and
hence p ,p ∈ I . Thus each summand
s(β) s(µ) ΣH(V)
s s s s = s p s s p s ∈ I .
α β∗ µ ν∗ α s(α) β∗ µ s(µ) ν∗ ΣH(V)
It follows that p ∈ I . Thus v ∈ ΣH(V), and hence E0 ⊂ ΣH(V). The reverse set
v ΣH(V)
inclusion is trivial. Thus ΣH(V) = E0.
Conversely, suppose that ΣH(V) = E0. To see that V is full, we need to show that
the ideal M∗M is all of L (E). For this, suppose that I is an ideal of L (E) containing
R R
MM∗. It suffices to show that L (E) = I: by Theorem 1, M∗M is an ideal of L (E)
R R
containing MM∗, and taking I = M∗M gives L (E) = M∗M, as needed.
R
By [15, Lemma 7.6], the subset H := {v ∈ E0: p ∈ I} of E0 is a saturated hereditary
I v
subset of E0. Since I contains MM∗, we have p ∈ I for all v ∈ V. Thus V ⊂ H ,
v I
and since H is a saturated hereditary subset, we get ΣH(V) ⊂ H . By assumption,
I I
ΣH(V) = E0, and now L (E) = I = I ⊂ I ⊂ I ⊂ L (E). So L (E) = I for
R E0 ΣH(V) HI R R
any ideal I containing MM∗. Thus V is full. (cid:3)
4. Contractible subgraphs of directed graphs
We start by stating the algebraic version of Crisp and Gow’s [10, Theorem 3.1]; for this
we need a few more definitions.
Let E be a directed graph. A finite path α = α α ...α in E with |α| ≥ 1 is a cycle if
1 2 |α|
s(α) = r(α) and s(α ) 6= s(α ) when i 6= j. Then E (respectively, a subgraph) is acyclic
i j
if it contains no cycles. An acyclic infinite path x = x x .... in E is a head if each r(x )
1 2 i
receives only x and each s(x ) emits only x .
i i i
IfE hasahead, wecangetanewgraphF bycollapsingtheheaddowntoasource. This
is an example of a desingularisation, and hence L (F) and L (E) are Morita equivalent
R R
by [1, Proposition 5.2]. Thus the “no heads” hypothesis in Theorem 3 below is not
restrictive.
MORITA EQUIVALENCES OF LEAVITT PATH ALGEBRAS 5
Theorem 3. Let R be a commutative ring with identity, let E be a directed graph with no
heads, and let {p ,s ,s } be a universal generating Leavitt E-family in L (E). Suppose
v e e∗ R
that G0 ⊂ E0 contains the singular vertices of E. Suppose also that the subgraph T of E
defined by
T0 := E0 \G0 and T1 := {e ∈ E1: s(e),r(e) ∈ T0}
is acyclic. Suppose that
(T1) each vertex in G0 is the range of at most one infinite path x ∈ E∞ such that
s(x ) ∈ T0 for all i ≥ 1.
i
Also suppose that for each y ∈ T∞,
(T2) there is a path from r(y) to a vertex in G0;
(T3) |s−1(r(y ))| = 1 for all i; and
i
(T4) e ∈ E1, s(e) = r(y) =⇒ |r−1(r(e))| < ∞.
Let G be the graph with vertex setG0 and one edge e for each β ∈ E∗\E0 with s(β),r(β) ∈
β
G0 and s(β ) ∈ T0 for 1 ≤ i < |β|, such that s(e ) = s(β) and r(e ) = r(β). Then L (G)
i β β R
is Morita equivalent to L (E).
R
In words, the new graph G of Theorem 3 is obtained by replacing each path β ∈ E∗
with s(β),r(β) in G0 of length at least 1 which passes through T by a single edge e ,
β
which has the same source and range as β. Note that the edges e in E with r(e) and s(e)
in G0 remain unchanged.
Let v ∈ E0. As in [10] define
B = {β ∈ E∗ \E0: r(β) = v,s(β) ∈ G0 and s(β ) ∈ T0 for 1 ≤ i < |β|}.
v i
Then B of E∗ corresponds to the set of edges G1 in G.
Sw∈G0 w
To prove Theorem 3, we apply Theorem 1 with V = G0 so that
M = span {s s : r(µ) ∈ G0}.
R µ ν∗
ThenM∗M isanidealofL (E)containingthesubalgebraMM∗,andM∗M andMM∗ are
R
Morita equivalent. We need to show that M∗M = L (E) and that MM∗ is isomorphic
R
to L (G). Our proof uses quite a few of the arguments from Crisp and Gow’s proof
R
of [10, Theorem 3.1]. In particular, Lemma 3.6 of [10] gives a Cuntz-Krieger G-family
in C∗(E), and since the proof is purely algebraic it also gives a Leavitt G-family in
L (E). Theuniversal propertyofL (G) thengives aunique homomorphism φ: L (G) →
R R R
L (E). Crisp and Gow used the gauge-invariant uniqueness theorem to show that their
R
C∗-homomorphism is one-to-one. The analogue here would be the graded uniqueness
theorem, however φ is not graded. Instead, to show φ is one-to-one, we adapt some clever
arguments from the proof of [1, Proposition 5.1] in Lemma 5 below which uses a reduction
theorem.
Theorem 4 (Reduction Theorem). Let R be a commutative ring with identity, let E be
a directed graph, and let {p ,s ,s } be a universal Leavitt E-family in L (E). Suppose
v e e∗ R
that 0 6= x ∈ L (E). There exist µ,ν ∈ E∗ such that either:
R
(1) for some v ∈ E0 and 0 6= r ∈ R we have 0 6= s xs = rp , or
µ∗ ν v
(2) there exist m,n ∈ Z with m ≤ n, r ∈ R, and a non-trivial cycle α ∈ E∗ such that
i
0 6= s xs = n r si . (If i is negative, then si := s|i|.)
µ∗ ν Pi=m i α α α∗
Proof. For Leavitt path algebras over a field this is proved in [3, Proposition 3.1]. We
checked carefully that the same proof works over a commutative ring R with identity. (cid:3)
6 L.O. CLARK,A. AN HUEF,AND P.LUITEN-APIRANA
Lemma 5. Let R be a commutative ring with identity. Let E and G be directed graphs,
and let φ: L (G) → L (E) be an R-algebra ∗-homomorphism. Denote by {p ,s ,s }
R R v e e∗
and {q ,t ,t } universal Leavitt E- and G-families in L (E) and L (G), respectively.
v e e∗ R R
Suppose that
(1) for all v ∈ G0, φ(q ) = p for some v′ ∈ E0; and
v v′
(2) for all e ∈ G1, φ(t ) = s for some β ∈ E∗ with |β| ≥ 1.
e β
Then φ is injective.
Proof. We follow an argument made in [1, Proposition 5.1]. Let x ∈ kerφ. Aiming for a
contradiction, suppose that x 6= 0. By Theorem 4 there exist µ,ν ∈ G∗ such that either
condition (1) or (2) of the theorem holds.
First suppose that (1) holds, that is, there exist v ∈ G0 and 0 6= r ∈ R such that
0 6= t xt = rq . Using assumption (1), there exists v′ ∈ E0 such that φ(q ) = p . Now
µ∗ ν v v v′
0 = φ(t xt ) = φ(rq ) = rφ(q ) = rp .
δ∗ γ v v v′
But rp 6= 0 for all 0 6= r ∈ R, giving a contradiction.
v′
Second suppose that (2) holds, that is, there exist m,n ∈ Z with m ≤ n, r ∈ R, and a
i
non-trivial cycle α ∈ E∗ such that 0 6= s xs = n r si . Since α is a non-trivial cycle,
µ∗ ν Pi=m i α
it has length at least 1. By assumption (2), φ(t ) = s where α′ is a path in E such that
α α′
|α′| ≥ |α| ≥ 1. Since φ is an R-algebra ∗-homomorphism we get
E G
n n n
0 = φ(t xt ) = φ r ti = r φ(t )i = r si .
µ∗ ν (cid:16)X i α(cid:17) X i α X i α′
i=m i=m i=m
Since |α′| = k for some k ≥ 1, s has grading k, and hence each si has grading ik. Thus
α′ α′
each term in the sum n r si is in a distinct graded component. But since s 6= 0,
we must have r = 0 foPria=lml i.i Tα′hus n r ti = 0, which is a contradiction. Inα′either
i Pi=m i α
case, we obtained a contradiction to the assumption that x 6= 0. Thus x = 0 and φ is
(cid:3)
injective.
Proof of Theorem 3. Let {p ,s ,s } be a universal Leavitt E-family in L (E). We apply
v e e∗ R
Theorem 1 with V = G0 to get a surjective Morita context between MM∗ and M∗M.
Since M and M∗ ⊂ L (E), we have M∗M ⊂ L (E). To see that L (E) ⊂ M∗M, let
R R R
s s ∈ L (E). We may assume that s(µ) = s(ν), for otherwise s s = 0. If s(µ) ∈ G0,
µ ν∗ R µ ν∗
then the Leavitt E-family relations give s s = s s s s ∈ M∗M and we are
µ ν∗ µ s(µ)∗ s(µ) ν∗
done. So suppose s(µ) ∈ T0. Then the graph-theoretic [10, Lemma 3.4(c)] implies that
B 6= ∅. Suppose first that B is finite. It then follows from the first part of [10,
s(µ) s(µ)
Lemma 3.6] that s(µ) is a nonsingular vertex. The second part of [10, Lemma 3.6] implies
that for any Cuntz-Krieger E-family {P ,S ,S } in C∗(E),
v e e∗
P = S S ;
s(µ) X β β∗
β∈Bs(µ)
the proof is purely algebraic and works for any Leavitt E-family in L (E). Thus
R
s s = s p s = s s s s = s s s s ∈ M∗M.
µ ν∗ µ s(µ) ν∗ X µ β β∗ ν∗ X µβ s(β)∗ s(β) (νβ)∗
β∈Bs(µ) β∈Bs(µ)
Next suppose that B is infinite. Since s(µ) ∈ T0 and B is infinite, the graph-
s(µ) s(µ)
theoretic [10, Lemma 3.4(d)] implies that there exists x ∈ T∞ such that s(µ) = r(x). By
MORITA EQUIVALENCES OF LEAVITT PATH ALGEBRAS 7
assumption (T2), there is a path α ∈ E∗ with r(α) ∈ G0 such that s(α) = r(x) = s(µ).
Now
s s = s p s = s s s s ∈ M∗M.
µ ν∗ µ s(µ) ν∗ µ α∗ α ν∗
Thus L (E) = M∗M. (We could have used Lemma 2 to prove that L (E) = M∗M, as
R R
Crisp and Gow do, but this seemed easier.)
Next we show that L (G) and M∗M are isomorphic. For v ∈ G0 and β ∈ ∪ B
R w∈G0 w
define
Q = p , T = s and T = s .
v v e β e∗ β∗
β β
Then {Q ,T ,T } is a Leavitt G-family in L (E); again this follows as in the proof of
v e e∗ R
[10, Theorem 3.1]. To see what is involved, we briefly step through this. Relations (L1)
follow immediately from the relations for {p ,s ,s }. To see that (L2) holds, let γ,β ∈
v e e∗
∪ B . Then T T = s s . By the graph-theoretic [10, Lemma 3.4(a)] neither γ nor
w∈G0 w e∗β eγ β∗ γ
β can be a proper extension of the other. Thus T T = s s = δ p = δ Q ,
e∗β eγ β∗ γ β,γ s(β) eβ,eγ s(eβ)
and (L2) holds.
To see that (L3) holds, let v ∈ G0 be a non-singular vertex. Then B is finite and
v
non-empty because it is equinumerous with r−1(v). Using the algebraic analogue of [10,
G
Lemma 3.6] again, we have
Q = p = s s = T T .
v v X β β∗ X eβ e∗β
β∈Bv eβ∈rG−1(v)
Thus (L3) holds and {Q ,T ,T } is a Leavitt G-family in L (E).
v e e∗ R
Now let {q ,t ,t } be a universal Leavitt G-family in L (G). The universal property
v e e∗ R
of L (G) gives a unique homomorphism φ: L (G) → L (E) such that for v ∈ G0,
R R R
β ∈ ∪ B ,
w∈G0 w
φ(q ) = Q = p , φ(t ) = T = s and φ(t ) = T = s .
v v v e e β e∗ e∗ β∗
β β β β
If v ∈ G0, then p = s s ∈ MM∗; if β ∈ B for some w ∈ G0, then r(β) ∈ G0, and
v v v∗ w
s = s s and s = s s ∈ MM∗. It follows that the range of φ is contained in
β β s(β)∗ β∗ s(β) β∗
MM∗. That φ is onto MM∗ again follows from work of Crisp and Gow. They take a
non-zero spanning element s s ∈ MM∗ and use the graph-theoretic [10, Lemma 3.4(b)],
µ ν∗
the algebraic [10, Lemma 3.6] and assumptions (T1)-(T4) to show that s s is in the
µ ν∗
range of φ. Thus φ is onto.
Finally, φ satisfies the hypotheses of Lemma 5, and hence is one-to-one. Thus φ is an
isomorphism of L (G) onto MM∗. (cid:3)
R
Remark 6. A version of Theorem 1 should hold for the Kumjian-Pask algebras associ-
ated to locally convex or finitely aligned k-graphs [6, 8]. But the challenge would be to
formulate an appropriate notion of contractible subgraph in that setting.
5. Examples
As mentioned in the introduction, the setting of Theorem 3 includes many known
examples. We found it helpful to see how some concrete examples fit.
Example 7. An infinite path x = x x ... in a directed graph is collapsible if x has no
1 2
exits except at r(x), the set r−1(r(x )) of edges is finite for every i, and r−1(r(x)) = {x }
i 1
(see [14, Chapter 5]). Consider the following row-finite directed graph E:
8 L.O. CLARK,A. AN HUEF,AND P.LUITEN-APIRANA
v v v v ...
1 2 3 4
v ...
0 x x x x
1 2 3 4
The infinite path x = x x ··· is collapsible. When we collapse x to the vertex v , as
1 2 0
described in [14, Proposition 5.2], we get the following graph F with infinite receiver v :
0
v v v
1 2 3
...
v
0
This fits the setting of Theorem 3: take G0 = {v : i ≥ 0}, and then T is the subgraph
i
defined by T0 = {s(x ): i ≥ 1} and T1 = {x : i ≥ 2}. Then T contains none of the
i i
singular vertices {v : i ≥ 2} of E, is acyclic, and satisfies the conditions (T1)–(T4). Thus
i
F is the graph G described in the theorem.
Example 8. Consider the directed graph F with source w and infinite receiver v:
∞
v w
An example of a Drinen-Tomforde desingularisation [12] of F is the row-finite graph E
with no sources below on the left: a head has been added at the source w of F and each
edge from w to v in F has been replaced with paths as shown. (This desingularisation is
also an example of an out-delay.) Then, since we are interested in Morita equivalence, we
delete the head at w to get the graph E below on the right.
v w ... v w
. .
. .
. .
Set T0 = E0\{v,w}andT1 = {e ∈ E1: s(e),r(e) ∈ T0}.Then thesubgraphT contains
none of the singularities of E, is acyclic, and satisfies conditions (T1)–(T4) of Theorem 3.
The graph F we started with is the graph G of Theorem 3.
Example 9. Consider again the graph F of Example 8. Label the infinitely many edges
from w to v by e for i ≥ 1. This time we will consider the in-delayed graph d (E) given
i s
MORITA EQUIVALENCES OF LEAVITT PATH ALGEBRAS 9
by the Drinen source-vector d : E0∪E1 → N∪{∞} (see [5, Section 4]) to be the function
s
defined by d (e ) = i−1 for i ≥ 1, d (v) = 0 and d (w) = ∞. Then the in-delayed graph
s i s s
d (E) given by d , as described in [5], is
s s
e
1
v0 w0
e
2
e
3
w1
w2
.
.
.
Now take T0 = d (E)0\{v0,w0}. Then T0 contains none of the singular vertices of d (E),
s s
and the corresponding subgraph T is acyclic. There are no infinite paths in d (E), and
s
hence conditions (T1)–(T4) of Theorem 3 hold trivially. The graph G of the theorem is
again the graph F that we started out with.
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[15] M. Tomforde. Leavitt path algebras with coefficients in a commutative ring. J. Pure Appl. Algebra
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(L.O.Clark,A.anHuef, P.Luiten-Apirana)Department of Mathematics and Statistics, Uni-
versity of Otago, P.O. Box 56, Dunedin 9054, New Zealand
E-mail address: lclark, astrid @maths.otago.ac.nz
E-mail address: [email protected]